How do you evaluate the limit #sin(5x)/sin(6x)# as x approaches #0#?

Question

Emma Aldridge · Accepted Answer

#5/6# #lim_(x to 0) sin(5x)/sin(6x)# we will use fundamental trig result: #lim_(z to 0) ( sin z)/z = 1# #= lim_(x to 0) (5(5sin(5x))/(5x))/(6(sin(6x))/(6x)) = 5/6#

Ezra Adams · Answer

undefined To evaluate the limit of sin(5x)/sin(6x) as x approaches 0, we can use the concept of limits and trigonometric identities. By applying the limit properties and the fact that sin(x)/x approaches 1 as x approaches 0, we can simplify the expression.

Using the limit property, we have:

lim(x→0) sin(5x)/sin(6x) = sin(5(0))/sin(6(0))

Since sin(0) is equal to 0, we have:

lim(x→0) sin(5x)/sin(6x) = 0/0

This is an indeterminate form, so we need to further simplify the expression. By using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite the expression as:

lim(x→0) sin(5x)/sin(6x) = lim(x→0) (5x)/(6x) * (sin(5x))/(sin(6x))

Now, we can cancel out the x terms:

lim(x→0) sin(5x)/sin(6x) = lim(x→0) 5/6 * (sin(5x))/(sin(6x))

Since sin(5x)/sin(6x) is still an indeterminate form, we can apply the limit property again:

lim(x→0) sin(5x)/sin(6x) = 5/6 * lim(x→0) (sin(5x))/(sin(6x))

Now, we can use the fact that sin(x)/x approaches 1 as x approaches 0:

lim(x→0) sin(5x)/sin(6x) = 5/6 * 1

Therefore, the limit of sin(5x)/sin(6x) as x approaches 0 is 5/6.